SIR B. C. BRODIE ON THE CALCULUS OE CHEMICAL OPERATIONS. 
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appear in the arrangement before that event, a,b,c... appear in the arrangement after 
that event, and where a,b,c... appear in the arrangement before that event, x,y,z . . . 
appear in the arrangement after that event, then the event is said to occur by the n 
substitutions of a for x, b for y, c for z, and these substitutions are termed the n causes 
of the event which concur for its production. 
The same remark is applicable here as in the case of an event occurring by a single 
“ cause,” namely, that as events may occur by a single “ cause” in any number of ways, 
so also may we have events occurring by a set of two or more “ causes ” in any number 
of ways, each set of “ causes ” representing an alternative hypothesis as to the mode of 
the occurrence of that event. 
(3) Definition: — 
If an event be regarded as occurring by the substitution of a for x , x is termed 
the symbol of the “ variable ” in that event, and a is termed a “ value ” of x (being that 
for which x is exchanged) ; if the event be regarded as occurring by the substitutions of 
a for x, b for y, c for z, x, y , z are termed the symbols of the “ variables ” in that event, 
and a , b, c the “ values ” of those “ variables,” and so on for any number of substitutions. 
(4) Definition : — 
A “constant weight” is a weight which, in any specified event or system of events, 
is not exchanged for any other weight. 
(5) It is a consequence of the above definitions that the “substitutions” by which 
events occur will be indicated to us by certain algebraical properties of the equations 
which express the results of those events. Thus, if an event occurs by the substitution 
of a for x, it is evident, from the definition, that the equation corresponding to the event 
must vanish when x is put equal to a ; and if the event occurs by one substitution, and 
in one way only by such a substitution, so that it can be referred to one cause alone, the 
equation will vanish under this condition, and under no other similar condition. If there 
be two alternative “ causes ” of the event, so that the event may be regarded as occur- 
ring either by the substitution of a for x, or of b for y , the equation will vanish when 
x is put equal to «, and also will vanish when y is put equal to b ; and if there be n 
alternative causes of the event, so that the event may occur either by the substitution of 
a for x , of b for y, of c for z, . . . the equation will vanish in n ways, namely, when either 
x is put equal to a, or y is put equal to b, or z is put equal to c, . . . If the event may 
be referred to two “ causes ” which concur for its production, so that it occurs by two 
substitutions, namely, the substitutions of a for x and of b for y, the equation will vanish 
if in that equation we simultaneously put x equal to a and y equal to b ; and if the event 
may be referred to n “ causes ” which concur for its production, so that it occurs by n 
substitutions of a for x , b for y, c for z . . ., the equation will vanish if in that equation 
we simultaneously put x equal to a, y equal to b, z equal to c . . . And, similarly, if the 
equation may not only be referred to n causes, but may be referred in rn ways to n 
causes, there will then be m ways in which the equation will vanish under a similar 
condition. 
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